§ DICTIONARY · CONCEPT

Fermat's principle

Light travels between two points along the path that takes the least time.

§ 01

Definition

Stated by Pierre de Fermat in 1662, this was the first variational principle in physics. From it alone — without any assumption about whether light is a particle or a wave — one can derive Snell's law of refraction, the straight-line propagation of light in a uniform medium, and the reflection law for mirrors.

Writing the total travel time as a functional of the path and demanding that it be stationary gives, at any interface between two media of speeds v₁ and v₂, the condition sin θ₁ / v₁ = sin θ₂ / v₂, which is Snell's law. Fermat's principle was the direct conceptual precursor of Maupertuis's principle of least action; it generalises, in modern optics, to the principle of stationary optical path length.

§ 02

History

Formulated by Pierre de Fermat in 1662 in reply to a challenge from the Cartesian physicist Claude Clerselier, defending the least-time derivation of Snell's law.